Pacific Journal of Mathematics
ALMOST DIFFEOMORPHISMS OF MANIFOLDS
RODOLFO DESAPIO
Vol. 26, No. 1 November 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 26, No. 1, 1968
ALMOST DIFFEOMORPHISMS OF MANIFOLDS
R. DE SAPIO
Let /: Mn —» Xn be a homotopy equivalence between two closed, differentiable (of class C°°) %-manifolds such that / n n induces, from the stable normal bundle Vk(X ) of X in the in + ifc)-sphere Sn+k, a bundle over Mn that is equivalent to the n n n+k stable normal bundle Vk(M ) of M in S . Then it is found that the disjoint union XnϋMn bounds a differentiable (n+1)- manifold Wn+ί with a retraction r: T7W+1 -» Xn such that the restriction r\Mn is equal to the given homotopy equivalence /. Furthermore, let n = 2q — 1 or 2g, where g ^ 3, and sup- pose that Xn is simply connected if n = 2g — 1, and that X™ is 2-connected if n — 2q. Then, if the restrictions of the n n+1 n+i bundles r*Vk(X ) and Vk(W ) to the (q - l)-skeleton of W are equivalent, where r*Vk(Xn) is the bundle induced by r: Wn+1 -» X" and F^(T^n+1) is the stable normal bundle of Wn+1 in *Sw+fc, then Mn and Xπ are diffeomorphic up to a point. In particular, Mn and Xw are homeomorphic.
It was S. P. Novikov who first gave sufficient conditions for homotopically equivalent, closed, simply connected ^-manifolds Mn and Xn to be homeomorphic, provided that n ^ 5. In this paper one of his sufficient conditions is replaced by a weaker condition. The other condition of Novikov is assumed here, namely that the given homotopy equivalence /: Mn —> Xn has the property that the induced bundle n n f*Vk(X ) is equivalent to Vk{M ). The homeomorphism obtained here is actually an almost diffeomor- phism (as was obtained by Novikov) in the following sense: an almost diffeomorphism of closed w-manifolds Mn and Xn is a homeomorphism g: Mn —> Xn that is a diffeomorphism except possibly at a single point. For n Φ 3,4 this is equivalent to the statement that there is a homotopy ^-sphere Vn such that Mn is diffeomorphic to the connected sum Xn # Vn. Of course Novikov's conclusion is stronger than the one obtained here in that he concludes that the homotopy ^-sphere Vn bounds a parallelizable manifold, which may not be the case here. This is due to the fact that we have removed one of Novikov's conditions on the homotopy equivalence. The reason for removing this condition is clear from the following example. Let n — 9 or 13 and let Mn be the tangent (n — l)/2-sphere bundle to the (n + l)/2-sphere. Let Xn be the connected sum of Mn with a homotopy ^-sphere that does not bound a parallelizable manifold; such homotopy ^-spheres exist at least for n = 9,13, and 15 (see [5]). Then Mn and Xn have the same homotopy type (they are almost
47 48 R. DE SAPIO diffeomorphic) and have trivial stable normal bundles in a high dimen- sional (n + &)-sphere Sn+k. But for any homotopy ^-sphere Vn that bounds a parallelizable manifold, the manifolds Mn and Xn # Vn are never diffeomorphic (see [2]). A more trivial example of this would be to take Mn = Sn the standard ^-sphere, and to take Xn to be a homotopy w-sphere that does not bound a parallelizable manifold, say for n = 9,13, or 15. In other words, what is desired is a necessary and sufficient condition for two closed ^-manifolds Mn and Xn of the same homotopy type to be almost diffeomorphic. A necessary condition is that there exists a homotopy equivalence /: Mn —> Xn such that / induces, from the stable normal bundle of Xn in Sn+k, a bundle over Mn that is equivalent to the stable normal bundle of Mn in Sn+k. This follows immediately from the fact that homotopy ^-spheres have trivial stable normal bundles in Sn+k (i.e. they are τr-manifolds). However, in general this necessary condition does not appear to be sufficient, although it is in fact a sufficient condition for almost diffeomorphism in either one of the following two cases: 1. n = 2q > 4 and M" is (q — l)-connected (see [6]). 2. n = 2q + 1, where q is even and Mn is (q — l)-connected and n stably parallelizable such that Hq(M ) is cyclic (see [2]). In general the necessary condition above does imply the existence of a cobordism between Mn and Xn with a certain nice property. Then in order to conclude that Mn and Xn are almost diffeomorphic we make an assumption about this cobordism. In all honesty, it must be said that this assumption is really quite unsatisfactory at least because it is not a necessary condition in general. Differentiate (or smooth) will mean of class C°° and all manifolds are assumed to be differentiate, compact, and oriented (except as noted).
1* The cobordism* For a space X and integer n ^ 0 there is defined the w-dimensional unoriented bordism group 3ln(X) as developed by Conner and Floyed [1]. The abelian group %ln(X) consists of bordism n n classes [ikP, f]2 of singular ^-manifolds (M , /), where M is a closed (possibly nonorientable) n-manifold and /: Mn —> X is a map. Suppose that Mn and Xn are closed ^-manifolds and that /: Mn —• Xn is a map. Then the singular ^-manifold (Mn,f) determines an n < n n n element [M ,f]ie Sίln(X ). Now the identity map id: X —>X also n n n determines an element [X , id]2 e ^ln{X ) and in general [M , f]2 Φ n [X , id]2. However, Conner and Floyd [1] give a necessary and n n sufficient condition for [M , f]2 = [X , id]2 in terms of the so called n n Whitney numbers of a map. If it is true that [M , f]2 = [X , id]2, then by definition there is a compact (n + l)-manifold Wn+\ possibly ALMOST DIFFEOMORPHISMS OF MANIFOLDS 49 nonorientable, with boundary the disjoint union Xn (J Mn and a map n 71 r: ψ»+i _> χ«> SUch that r | M = / and r | X is the identity on X*. That is, r is a retraction of Wn+1 onto Xn. n n n We have a map /: M —>X of closed ^-manifolds. Let μn e Hn(M ; Z2) n n be the fundamental class. Then for any 7 e H (M ; Z2) there is defined e the Kronecker index
μnyeZ2 is defined. Following [1], this integer mod 2 is called a m m n Whitney number of the map f associated with c e H (X : Z2). We remark that the Whitney numbers are also defined when Mn is non- orien table in exactly the same way. There are also defined the Whitney numbers of the map id: Xn —*Xn. Now a result of [1, Th. n n n n 17.2, p. 47] asserts that [M , f]2 = [X , id\ if and only if /: M -> X and id: Xn —> Xn have the same Whitney numbers.
NOTATION. Let Vk(X) denote the stable normal bundle of a manifold X in euclidean space.
n n LEMMA 1. Let f: M —> X be a homotopy equivalence such that n n the induced bundle f*Vk{X ) is equivalent to Vk(M ). Then there is an (n + l)-manifold Wn+1 with boundary the disjoint union Xn U Mn and a retraction r: Wn+1 -> Xn such that r\Mn = f.
n n Proof. Since f*Vk(X ) is equivalent to Vk(M ),f induces the stable tangent bundle of Mn from the stable tangent bundle of Xn. Hence by the naturality of the Stiefel-Whitney classes we have
f*(Wi) = Wi , where ifii is the ί-th Stiefel-Whitney class of X71. By previous remarks it is sufficient to show that /: Mn —> Xn and id: Xn-+Xn have the same n n Whitney numbers, as follows. If μn e Hn(M ; Z2) and μn e Hn(X ; Z2) are the fundamental classes, then
m m n since / is a homotopy equivalence and hence, given c £ H (X ; Z2) and iι + + ik — n — m, we have
m m 2* The condition* THEOREM. Let f: Mn —> Xn be a homotopy equivalence of simply n connected, closed n-manifolds such that f*Vk(X ) is equivalent to n Vk(M ). Then: (a) The disjoint union Mn U Xn bounds an (n + l)-manifold Wn+1 with a retraction r: Wn+1 -> Xn such that r\Mn = f. (b) Let r: Wn+1 —» Xn be as in (a) and suppose that the restric- n n+1 tions of the bundles r*Vk(X ) and Vk(W ) to the [(n - l)/2]-skeleton of Wn+1 are equivalent. Then Mn and Xn are almost diffeomorphic for n ^ 5, provided that Xn is 2-connected when n is even. REMARK. Here [x], where x is a real number, denotes the greatest integer less than or equal to x. Furthermore, if H*(Xn) has torsion consisting entirely of elements of order two, then, by considering the Pontrjagin numbers of the map f: Mn —* Xn (cf. [1, p. 48]), we may apply [1, Th. 17.5, p. 49] to conclude that the manifold Wn+ι of Lemma 1 (and hence of (a) in the theorem) may be taken to be orientable; hence by application of the method of Browder and Novikov as employed in the following proof, the manifold Wn+1 of (a) in the theorem may be taken to be simply connected. Proof. For convenience write n = 2q — 1 or 2g, q ^> 3. In the n n+1 first place if r*Vk(X ) is equivalent to Vk(W ), then it can be shown that this condition is equivalent to the condition of Novikov and hence it follows from the theorem of Novikov that M* and Xn are almost diffeomorphic. However, the assumptions made here are sufficient to insure that we may apply the technique of Browder and Novikov (see [6] or [8]) to kill the kernel (which is finitely generated at each stage) n+1 n of r*: π{(W ) -^πi(X ) in dimensions i < q by spherical modifications n+1 of w . It is only necessary to check that each element in the kernel of r* for i < q is represented by a smooth embedding S* —• Wn+1 with trivial normal bundle. The existence of the embedding is due to Whitney-Haefliger (since n + 1 > 2i + 1 for i < q) and the fact that it has trivial normal bundle follows from an argument similar to one due to Milnor [7, Th. 3] (the assumption concerning the (q — l)-skeleton n+1 n n+1 of W plays a role here to conclude that r*Vk{X ) and Vk(W ) are equivalent when restricted to an open submanifold of Wn+1 that contains the (q — l)-skeleton of Wn+1). Hence we may assume that r*: πi(Wn+1) —+ ^(Jf*) is an isomorphism for i < q and therefore, from considering the exact homotopy sequence of the pair (Wn+1, Xn), it rw+1 n n follows that 7Γί(W , X ) = 0 for i < q. Since r | M is a homotopy equivalence it follows similarly that πi(Wn+1, Mn) = 0 for i < q. It follows from handlebody theory that WnJrl may be obtained ALMOST DIFFEOMORPHISMS OF MANIFOLDS 51 from Xn (or Mn) by successively attaching a finite number of "handles" to Xn x I (or Mn x /, where I = [0,1] is the unit interval) or, what is the same thing, by performing a finite sequence of spherical modi- fications on Xn (or Mn). The argument now splits up into two cases, n — 2q — 1 and n — 2q. Both arguments use the fact that +\ Xn) - πi(Wn+\ Mn) = 0 , % Case 1. n = 2q - 1. It follows from (1) and [9, p. 539] that Wn+1 is diffeomorphic to (Xn x I) U U #5 x D ψ \i=i where 9?: (J}=i (3-Dj x #g) —> Xw x 1 is an embedding of s copies of dDq x Dq = S9"1 x Dq (Dq the unit #-disc in real g-space) disjointly in Xn x 1 and where the handles D) x Dq are attached along Xn x 1 by means of φ. Now in the portion of the exact homotopy sequence n +1 πq(W*+\ X ) — πq^X») -^U πq^(W* ) where i: Xn —• PF%+1 is the inclusion and ί^ is an isomorphism, we see that d is trivial and hence φ \ dDq x 0 is homotopically trivial. Thus φ is homotopically trivial and, since Xn is simply connected, it follows from the engulfing theorem of [4] that φ(\J3- dD] x 0) is contained in the interior of a combinatorial %-disc embedded in Xn x 1. The boundary of this disc may be smoothed in Xn x 1 to conclude that φ(\Jj 3D) x Dq) is contained in a smooth n-disc in Xn x 1. Hence Wn+ί is diffeomorphic to the boundary connected sum {Xn x I, n n+1 n+1 n+1 n+1 q X x 1) # (W2 , dW2 ), where W2 = D \Jφ (U5=i D) x D ). That n+ί n+1 n is, W2 is a handlebody such that 3W2 #X is deffeomorphic to n n+ί M . It is well known that dW2 is a simply connected ^-manifold w n+1 and since X and Λf^ have the same homotopy type, Hi(dW2 ) = 0 +1 for 0 < i < w. Thus dW2 is a homotopy ^-sphere, as desired. REMARK. I should like to thank M. Hirsch for informing me that the engulfing theorem of [4] could be applied here. I later discovered that this engulfing theorem could also be applied to the more com- plicated situation found in Case 2 below. Case 2. n = 2q. It follows from (1) and [9] that there is a smooth embedding φ\ U (&D5 x Dq+ι) > Γxl that (by the same argument as that of Case 1) embeds Ui=i 3-D? x Dq+1 52 R. DE SAPIO into the interior of an w-disc Dn that is embedded in Pxl, and there is a smooth embedding ψ: U (dDl+1 x Dq) > (Xn x 1) # dW?+1 , n+1 w+1 x m n+1 where TF2 = 7) \Jφ (U5=i D) -D )> such that W is diffeomor- phic to the boundary connected sum n n w+1 +1 +1 [(X x /, X x 1) # (TΓa , 3W? )] U Γύ D\ x ψ Li=l with the handles 7)ffl x Dg attached by means of ψ. We shall show +1 q %+1 that we may assume that ψ embeds \JidDl x D into 3TΓ2 -point, from which it will follow that Wn+ι is diffeomorphic to (Xn x 7, Γxl)ϊ (JVn+1, dNn+ι), where 1 W+1 ?+1 iv^ = TF2 u (U A x - and hence Mn is diίfeomorphic to Xn#dNn+1; but, as in Case 1, dNn+1 is a homotopy w-sphere. LEMMA 2. 7/ ^4. and B are simplicial complexes such that A is simply connected and B is (q — ϊ)-connected, then ΊZ^A V B) = π{(A) + πt(B) for l^i^q. This is a well known result in homotopy theory and follows from the fundamental fact that τr,(A V B) = πi(A) + π,(B) + πi+1(A x B,AV B) (ί > 1) and the relative Hurewicz theorem, noting that Hi+1(A x B, A V JB) = 0 for i m q. We shall apply this lemma to the case where n %+1 A = X x 1-point , £ = 3ΐF2 -point . Since A V B has the homotopy type of (Xn x 1) fydW^1 with a point removed, we shall write for convenience % w+1 A V B - [(X x 1) # 3TΓ2 ]-point . We begin by showing that we can assume that r | A V B is the identity on A — Xn x 1-point and is, after changing the g-handles in the decomposition of Wn+1 if necessary, homotopically trivial on B = SW^-point. It will then follow that the homomorphism ( 2 ) πg(A) + πq(B) - πq(A V B) > πq(A) that is induced by r \ A V B is the identity on τrq(A) and is zero on ALMOST DIFFEOMORPHISMS OF MANIFOLDS 53 πq(B). We shall divide this argument into two steps. For convenience introduce the notation VΓ+1 = (Xn x /) U then dV?+1 is the disjoint union of the "lower" boundary Xn = Xn x 0 n Λ+1 and the "upper" boundary Y» = (X x 1) #3TF2 . Step 1. Consider the portion of the exact homotopy sequence of the pair (VΓ+\ Xn): 1 n o —> πq(x*) τ==i πq( vr ) τ=i **( vr\ x ) —> o, r* θ where h* is induced by the inclusion. Since r is a retraction, this short exact sequence splits and θ is the unique homomorphism such that r*θ is zero and h^θ is the identity. By the relative Hurewicz theorem the g-cells ΰ] x 0 attached by means of φ represent a set of n free generators Xj(j = 1, , s) for πq(V^\ X ). Let 3,. = 0(3,-) O' = 1, .--,8) and note that, by the theorems of Haefliger, each zά may be represented by a g-sphere S) that is smoothly embedded in the interior of FΓfl. We may suppose that the S) are pairwise disjoint and disjoint from Xn x [0, 1/2], Now choose an %-disc that is smoothly embedded in Xn x I and then, for each j = 1, , s, choose a smooth (q — l)-sphere within this w-disc in Xn x J such that these (q — l)-spheres are pairwise disjoint. For each j = 1, « ,s run a tube of the form Sq~ι x 7 in the complement of Xn x [0, 1/2] in V^1 from the j-th (q — l)-sphere in Xn x J to S% these tubes being pairwise disjoint, such that the result is the manifold I* x -| with g-cells attached. We may then thicken these g-cells so that the result is a manifold of the form n+1 n +ί V2 - (X x [0, 1/2]) U (U D) x D« λ \j = l that is smoothly embedded in Vΐ+1; here λ is a smooth embedding of [JU^D'j x Dq+1 into an ^-disc in Xn x J. Now let B] c Dn+1 czXn x [0, 1/2] be a g-dise with dBj = λ(3J9? x 0) such that B) \Jλ {D) x 0) is a (/-sphere embedded in 54 R. DE SAPIO λ Clearly, we may suppose that the g-sphere B] \Jλ {D) x 0) represents +ι the element z3- in πq{V? ). But then z3- = 0 q and so τ\B ό\Jλ {D) x 0) is homotopically trivial for each j and hence n+1 %+1 r I W3 -point is homotopically trivial. In particular, r \ d W3 -point is homotopically trivial. Step 2. We shall now show that we can isotopically deform the n+1 + +1 manifold V2 onto V? \ the isotopy lying within Vΐ and leaving fixed a neighborhood of Xn x 0. It will then follow from Step 1 that, by replacing W?+1 by W^+ι if necessary, we can assume that r | A V B +1 ι +1 ni is homotopically trivial on B = d T7<Γ -point. The closure C( VΓ - V2 ') of the complement of F*+1 in FΓ+1 is an (n + l)-manifold with simply connected "lower" boundary Y? = (Xn x lβ)$dWz+ι and simply con- n n+1 nected "upper" boundary Γ* = (X x 1) #dW2 . It is enough to show that C1(FΓ+1 - V^1) is diffeomorphic to the product Y? x /; by the Λ,-cobordism theorem of Smale this is equivalent to showing that +1 n+1 n Cl( FΓ - V2 ) is an Λ-cobordism between Y2 and Yf. Hence we +1 n+1 must show that the groups J3i(Cl(FΓ — V2 ), Yk) are zero for all i r +1 %+1 and k = 1, 2. Now by excision, JH i(Cl(FΓ - F2 ), Y2) is isomorphic w+1 %+1 to iϊί(V1 , F2 ) and thus is zero if we show that the inclusion α. yn+iς-.yn+i jnc[Uces isomorphisms of homology in all dimensions. The inclusion a induces a map of split short exact sequences of homology groups: o —> Hάxη ~^ Hά vrι) ^=ί Hi( vr\ xn) —> o ^=n Htivr1) τ=^ H^vr1, xη —> o r* θ The maps i*, r*, h*, θ of this commutative diagram are denoted by the same symbols as those used in the homotopy sequence of (V?+1, X") appearing above, and no confusion should arise because of this. Now n+ι +1 n Q V2 and VΓ each have the homotopy type of X V (Vs S j) and n+1 1 n n 71 a: V2 cz V^ is the identity on X = X x 0. Hence a*: H^X ) —> n +1 n 1 n Ht(X ) is the identity for all i and Hi(V? , X ) = H^V^ , X ) = 0 for all i Φ q. In dimension i = q we recall the set of free generators +1 {xj I j Φ 1, . , s} for fl"g(FΓ , X") and that the element ALMOST DIFFEOMORPHISMS OF MANIFOLDS 55 q q was represented by the smoothly embedded g-sphere B \Jλ {D ό x 0) +1 q n+1 in VΓ . But B)\Jλ(D ά x 0) also represents an element z'j e Hq(V2 ) such that a*(Zj) — zά. Furthermore, the elements h*(z'j) (j = 1, , s) n+ n form a set of free generators for Hq(V2 \ X ) such that betv/een Y2 and F*, as claimed. We now turn to the study of the embedding ψ. Note that +1 +l ψ I dD\ x 0 represents an element in πq(A\/B) and clearly rψ \ dD\ x 0 is homotopically trivial. Therefore, since the homomorphism of (2) is +ι the identity on πq(A) and zero on πq(B), ψ \ dDl x 0 is homotopic to a map dDl+1 x 0-+J3 and thus ψ is homotopic to a map U (3DΓ1 x 0) > B = 3^+1-point . ΐ = l We should again like to apply the engulfing theorem of Hirsch and Zeeman [4] and conclude that ψ is isotopic to an embedding with image in B. To this end note that B is a (q — l)-connected 2g-manifold and hence has the homotopy type of a wedge of g-spheres \f S q k k embedded in S, each ^-sphere Si smoothly embedded in B. Hence ψ q is homotopic to a map with image in y k S k and it follows that in 3 ?+1 the language of [4], t(U; A x 0) is (Vfc Sl)-inessential. Since q Avΰis 2-connected and π^A V B, yk S k) = 0 for i g 2 (recall that A = Xn x 1-point is 2-connected by hypothesis) all of the conditions +ι of the engulfing theorem [4] are satisfied and hence ψ(\J{ dD\ x 0) is contained in a regular neighborhood of V * Si in n n+1 A V B = [(X x 1) # 3TΓ2 ]-point . It is well known that the boundary of this regular neighborhood may +1 be smoothed in A V B and hence we can assume that ΊHLJ; ^? X 0) q is contained in a smooth regular neighborhood of \f k S k in A V B. Now w+1 +1 d TF2 -interior of smooth w-dise c d W^Γ -point = B is a smooth regular neighborhood of VA Si in B and hence in A V B. But a result of [3, Th. 1] asserts that any smooth regular neighbor- hood may be isotopically deformed onto any other, and hence Ψ(\Ji dDl+1 x 0) may be isotopically deformed into B, completing the 56 R. DE SAPIO proof of the theorem. n n COROLLARY. Let f: M —> X be a homotopy equivalence of simply n n connected n-manifolds such that f*Vk(X ) is equivalent to Vk(M ). Then: (a) The disjoint union Xn U Mn bounds a {n + l)-manifold Wn+1 such that there is a retraction r: Wn+1 —• Xn with r \ Mn = /. n+ι ίl+1 n (b) Let W be as in (a) and suppose that τri(TF , X ) = 0 for i ^ [(n — l)/2]. Let n ^ 5 and if n is even, then assume that Xn is 2-connected. Then Mn and Xn are almost diffeomorphic. The corollary follows from the proof of the theorem. REMARK. It can be shown that any homotopy ^-sphere bounds a simply connected (n + l)-manifold Nn+ί. In view of the above theorem the following question arises naturally: Can JV^1 be taken to be [(n — l)/2]-connected? If the answer is yes, then the condition on Wn+1 given in (b) of the theorem is necessary and sufficient for Mn and Xn to be almost diffeomorphic (n ^ 5 here). However, the answer is negative for n = 2q — 1 and q = 3, 5, 6, 7 (mod 8) since in this case Nn+1 would necessarily be paralleliz- able if it were (n — l)/2-connected. This is so because τrg_1(>SO) = 0 for q = 3, 5, 6, 7 (mod 8). Hence the homotopy sphere obtained in (b) of the theorem bounds a parallelizable manifold for n — 2q — 1 and q = 3, 5, 6, 7 (mod 8), and thus the condition given there is not in general necessary for Mn and Xn to be almost diffeomorphic. REFERENCES 1. P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete 33, Springer, Berlin-Gottingen-Heidelberg, 1964. 2. R. De Sapio, Actions of θik+u Michigan Math. J. 14 (1967), 97-100. 3. M. W. Hirsch, Smooth regular neighborhoods, Ann. of Math. 76 (1962), 524-530. 4. M. W. Hirsch and E. C. Zeeman, Engulfing, Bull. Amer. Math. Soc. 72 (1966), 113-115. 5. M. Kervaire and J. Milnor, Groups of homotopy spheres: I, Ann. of Math. 77 (1963), 503-537. 6. R. K. Lashof, Some theorems of Browder and Novikov on homotopy equivalent manifolds (mimeographed notes), The University of Chicago. 7. J. Milnor, A procedure for killing homotopy groups of differentiable manifolds, Proceedings of Symposia in Pure Math., A.M.S., Vol. Ill, 1961, 39-55. 8. C. T. C. Wall, An extension of results of Novikov and Browder, Amer. J. Math. 88 (1966), 20-32. 9. A. H. Wallace, A geometric method in differential topology, Bull. Amer. Math. Soc. 68 (1962), 533-542. Received March 7, 1967. The preparation of this paper was sponsored in part by NSF Grant GP-5860. UNIVERSITY OF CALIFORNIA, LOS ANGELES PACIFIC JOURNAL OF MATHEMATICS EDITORS H. ROYDEN J. DUGUNDJI Stanford University Department of Mathematics Stanford, California University of Southern California Los Angeles, California 90007 R. R. PHELPS RICHARD ARENS University of Washington University of California Seattle, Washington 98105 Los Angeles, California 90024 ASSOCIATE EDITORS E. F. BECKENBACH B. H. NEUMANN F. WOLF K. YOSIDA SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA STANFORD UNIVERSITY CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF TOKYO UNIVERSITY OF CALIFORNIA UNIVERSITY OF UTAH MONTANA STATE UNIVERSITY WASHINGTON STATE UNIVERSITY UNIVERSITY OF NEVADA UNIVERSITY OF WASHINGTON NEW MEXICO STATE UNIVERSITY * * * OREGON STATE UNIVERSITY AMERICAN MATHEMATICAL SOCIETY UNIVERSITY OF OREGON CHEVRON RESEARCH CORPORATION OSAKA UNIVERSITY TRW SYSTEMS UNIVERSITY OF SOUTHERN CALIFORNIA NAVAL WEAPONS CENTER Printed in Japan by International Academic Printing Co., Ltd., Tokyo, Japan aicJunlo ahmtc 98Vl 6 o 1 No. 26, Vol. 1968 Mathematics of Journal Pacific Pacific Journal of Mathematics Vol. 26, No. 1 November, 1968 Efraim Pacillas Armendariz, Closure properties in radical theory ...... 1 Friedrich-Wilhelm Bauer, Postnikov-decompositions of functors ...... 9 Thomas Ru-Wen Chow, The equivalence of group invariant positive definite functions ...... 25 Thomas Allan Cootz, A maximum principle and geometric properties of level sets ...... 39 Rodolfo DeSapio, Almost diffeomorphisms of manifolds ...... 47 R. L. Duncan, Some continuity properties of the Schnirelmann density...... 57 Ralph Jasper Faudree, Jr., Automorphism groups of finite subgroups of division rings ...... 59 Thomas Alastair Gillespie, An invariant subspace theorem of J. Feldman...... 67 George Isaac Glauberman and John Griggs Thompson, Weakly closed direct factors of Sylow subgroups ...... 73 Hiroshi Haruki, On inequalities generalizing a Pythagorean functional equation and Jensen’s functional equation ...... 85 David Wilson Henderson, D-dimension. I. A new transfinite dimension. . . . . 91 David Wilson Henderson, D-dimension. II. Separable spaces and compactifications ...... 109 Julien O. Hennefeld, A note on the Arens products ...... 115 Richard Vincent Kadison, Strong continuity of operator functions...... 121 J. G. Kalbfleisch and Ralph Gordon Stanton, Maximal and minimal coverings of (k − 1)-tuples by k-tuples ...... 131 Franklin Lowenthal, On generating subgroups of the Moebius group by pairs of infinitesimal transformations...... 141 Michael Barry Marcus, Gaussian processes with stationary increments possessing discontinuous sample paths ...... 149 Zalman Rubinstein, On a problem of Ilyeff ...... 159 Bernard Russo, Unimodular contractions in Hilbert space ...... 163 David Lee Skoug, Generalized Ilstow and Feynman integrals...... 171 William Charles Waterhouse, Dual groups of vector spaces ...... 193